Scaling for stress similarity and distorted-shape similarity in bending and torsion under maximal muscle forces concurs with geometric similarity among different-sized animals

Abstract

When geometric similarity, or isometry, prevails among animals of different sizes their form and proportions are similar. Weight increases as the cube of the length dimension, while cross-sectional areas increase as its square, so in load-bearing structural elements the stress, caused by the body weight, increases in direct proportion to the length dimension, both for pure axial loads and for transverse bending and torsional loads. On this account, large body sizes would be expected to set up compensatory selection on the proportions of supporting structures, making them disproportionately thicker as required to maintain similar, size-independent safety factors against breakage. Most previous scaling theories have assumed that the strength of support elements has evolved with respect to loads due to the body weight. But then, from the arguments above, a scaling principle different from the geometric similarity rule would be required in order for safety factors to remain similar among different-sized animals. Still, most comparable animals of 'similar kind' scale in accordance with the geometric similarity rule. Here, we instead argue that muscle forces cause much larger loads on structural support elements during maximum performance events (such as during prey capture or escape from predators) than do loads dictated by the body weight (such as during cruising locomotion), and that structural strength therefore might evolve with respect to maximal muscle forces rather than to the body weight. We explore how the transverse and longitudinal lengths of structural support elements must scale to one another, and to muscle transverse length, in order to satisfy each of the following, functionally based, similarity principles for support elements placed in bending, or in torsion, by maximal muscle forces during locomotion: (1) similarity in axial stress, or (2) in torsional shear stress, and (3) similarity in bent shape, or (4) in twisted shape. A dimensional relationship that satisfies all four conditions actually turns out to be the geometric similarity rule. These functional attributes may therefore help to explain the prevalence of geometric similarity among animals. Conformance of different-sized species with the geometric similarity principle has not been directly selected for as such, of course, but may have arisen as a by-product of adaptation in morphological proportions, following upon selection, in each separate species-lineage, for adequate and similar safety factors against breakage, and similar optimal distorted shapes, of structural support elements placed in bending, or in torsion, by maximal muscle forces.